Abstract: The complex Hilbert space has been an axiom of every quantum formulation since Dirac and von Neumann. No physical theory has explained why the universe uses complex numbers rather than real numbers, quaternions, or any other algebraic structure. Within the Algorithmic Theory of Reality (ATR) framework, we show that the complex field is the unique choice that keeps the rendering engine's entropy growth rate below the observer's Zeno Threshold (Zα) while maintaining reversibility and composability. Real Hilbert spaces fail because bipartite states contain hidden parameters (e.g., σy ⊗ σy) inaccessible to local measurements. Tracking these requires an additional Landauer cost, causing the entropy rate to scale as O(K4) and immediately breach the observer's Zα, triggering premature wavefunction collapse. Quaternionic Hilbert spaces fail because non-commutativity prevents the construction of a well-defined tensor product, breaking memory aliasing. The complex field is the thermodynamic optimum: it provides exactly the right number of state-space parameters for perfect local tomography at zero excess Landauer cost. This result promotes ATR's Axiom 1 (complex Hilbert space) from an assumption to a theorem. Computational verification suite available at: github.com/srdrymn/atr-zeno-hilbert-space-derivation
Serdar Hanzala Yaman (Thu,) studied this question.